Answer to Question #140549 in Real Analysis for Nidhi

Question #140549
Q.1 examine the convergence of series: 3×4/5^2 +'5×6/7^2 + 7×8/9^2 + ......
1
Expert's answer
2020-10-27T19:06:39-0400

"\\frac{3.4}{5^2}+\\frac{5.6}{7^2}+\\frac{7.8}{9^2}+\\dots = \\sum_{n=2}^\\infin\\frac{(2n-1).(2n)}{(2n+1)^2}= \\sum_{n=2}^\\infin U_n"

Then we have

"\\lim_{n\\to \\infin} U_n= \\lim_{n\\to\\infin}\\frac{(2n-1).(2n)}{(2n+1)^2} = \\lim_{n\\to\\infin}\\frac{(2-\\frac{1}{n}).2}{(2+\\frac{1}{n})^2}= \\frac{2.2}{4}=\\frac{4}{4}=1 \\ne0"

Since "\\lim_{n\\to \\infin}U_n\\ne0" then by Cauchy's test for divergence the series is divergent(not convergent)


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